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A357034 a(n) is the smallest number with exactly n divisors that are hoax numbers (A019506).

%I A357034 #12 Oct 14 2022 08:52:59
%S A357034 1,22,308,638,3696,4212,18480,26400,55080,52800,73920,108108,220320,
%T A357034 216216,275400,324324,432432,550800,734400,1908000,1144800,1101600,
%U A357034 1377000,1652400,3027024,2203200,4039200,2754000,3304800,5724000,6528600,9180000,8586000,5508000
%N A357034 a(n) is the smallest number with exactly n divisors that are hoax numbers (A019506).
%e A357034 1 has no divisors in A019506, so a(0) = 1;
%e A357034 22 has divisors 1, 2, 11, 22, and 22 = A019506(1), so a(1) = 22.
%e A357034 308 has divisors 1, 2, 4, 7, 11, 14, 22, 28, 44, 77, 154, 308 and 22 = A019506(1), 308 = A019506(14), so a(2) = 308.
%t A357034 digitSum[n_] := Total @ IntegerDigits[n]; hoaxQ[n_] := CompositeQ[n] && Total[digitSum /@ FactorInteger[n][[;; , 1]]] == digitSum[n]; f[n_] := DivisorSum[n, 1 &, hoaxQ[#] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[10, 10^5] (* _Amiram Eldar_, Sep 26 2022 *)
%o A357034 (Magma) hoax:=func<n| not IsPrime(n) and (&+Intseq(n, 10) eq &+[ &+Intseq(p, 10): p in PrimeDivisors(n)])>; a:=[]; for n in [0..33] do k:=1; while #[d:d in Set(Divisors(k)) diff {1}|hoax(d)] ne n do k:=k+1; end while; Append(~a,k); end for; a;
%Y A357034 Cf. A019506.
%K A357034 nonn,base
%O A357034 0,2
%A A357034 _Marius A. Burtea_, Sep 20 2022